Cotangent cohomology and higher-order obstructions of matroids

Abstract

Matroid Stanley-Reisner rings admit essential deformation obstructions of arbitrarily high order over every field, already at corank three. For each $m\ge 3$, we construct a connected cosimple matroid and $m$ fine-homogeneous first-order classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. A specified joint $(m-1)$-jet nevertheless has no lift to order $m$, even without preserving the grading. The construction rests on formulas for every multigraded component of the second cotangent cohomology $T^2$ and of its primary cup image, expressed through flats and parallel classes of the dual matroid. These formulas show that $T^2$ vanishes exactly when every connected component has corank at most two, or equivalently when there is no $U_{1,4}$-minor. This proves a conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello (2025) and characterizes formal smoothness of the affine deformation functor. For the duals of rank-three truncations of $U_{2,m}\oplus U_{2,n}$, $m,n\ge 3$, each nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and the quotient has an explicit presentation with $m+n-2$ minimal homogeneous generators.

arXiv comment63 pages. Supplementary verification code and finite certificates included. Revised exposition, references and declarations; main results unchanged. An earlier version (Zenodo v3) was published on 23 September 2026: https://doi.org/10.5281/zenodo.22907302